An image interaction approach to quantum-phase engineering of two-dimensional materials

Tuning electrical, optical, and thermal material properties is central for engineering and understanding solid-state systems. In this scenario, atomically thin materials are appealing because of their sensitivity to electric and magnetic gating, as well as to interlayer hybridization. Here, we introduce a radically different approach to material engineering relying on the image interaction experienced by electrons in a two-dimensional material when placed in proximity of an electrically neutral structure. We theoretically show that electrons in a semiconductor atomic layer acquire a quantum phase resulting from the image potential induced by the presence of a neighboring periodic array of conducting ribbons, which in turn modifies the optical, electrical, and thermal properties of the monolayer, giving rise to additional interband optical absorption, plasmon hybridization, and metal-insulator transitions. Beyond its fundamental interest, material engineering based on the image interaction represents a disruptive approach to tailor the properties of atomic layers for application in nanodevices.

We consider a semiconductor monolayer under the conditions described in the main text, characterized by a parabolic conduction band of effective mass m * , partially filled to a Fermi level E 0 F . Conduction electrons can then be labeled by the in-plane wave vector k = (k x , k y ), such that the electron energies are ε k = 2 k 2 /2m * relative to the bottom of the conduction band. We also incorporate a periodic image potential produced by interaction with a neighboring conductive ribbon array of parameters and material composition as described in the main text (see Fig. 1b in the main text). The image potential landscape is introduced through a term V im (x) = −V 0 p(x) in the electron Hamiltonian, where V 0 > 0 measures the magnitude of the image interaction and p(x) = ∞ =−∞ θ( a + b − x)θ(x − a) follows the profile of the ribbon array (width b, period a) by means of step functions. Electron bands are then emerging, so we restrict k x to the 1BZ and introduce a band index n to label electron states by (k , n) with |k x | < π/a. We only consider in-plane electron motion (i.e., in the R = (x, y) plane), under the assumption that electron states are tightly confined along the out-of-plane direction, such that their wave functions can be factorized as ψ k n (R)ψ ⊥ (z), where ψ ⊥ (z) is shared by all states. In addition, we approximate the out-of-plane probability density as |ψ ⊥ (z)| 2 ≈ δ(z) (i.e., the 2D material limit). The remaining in-plane components are governed by the Schrödinger equation with the Hamiltonian where H 0 (R) = − 2 ∇ 2 R /2m * and we introduce electron Coulomb repulsion through the Hartree potential 1 V H (R) = e 2 d 2 R n(R ) − n 0 /|R − R |, where n(R) = 2 k n f k n |ψ k n (R)| 2 is the electron density, the factor of 2 accounts for spin degeneracy, and f k n is the Fermi-Dirac distribution. Here, we calculate the electronic band structure at zero temperature, such that f k n = θ(E F − ε k n ), where the Fermi energy E F is adjusted to make the average electron density equal to that of the unperturbed 2D semiconductor n 0 . Then, E F depends on the applied image potential and generally differs from E 0 F . We solve Eq. (1) iteratively by calculating the Hartree potential at each step, fixing E F to preserve the average electron density, and mixing the new Hartree potential with the previous one until convergence is achieved after a few iterations.
Because of the periodicity of the Hamiltonian (i.e., H(R) = H(R + ax) for any integer ), the eigenstates can be written as ψ k n (R) = e ik ·R u k n (R)/L (Bloch's theorem), where L 2 is the semiconductor area, and the functions u k n (R) also satisfy u k n (R + x) = u k n (R) for any integer . In addition, since the image potential only depends on x, we can factorize the wave functions and separate the energies as ψ k n (R) = e ik ·R u kxn (x)/L, (3a) where we have plane waves in the direction of translational invariance y. Along the direction of periodicity x, we find ε x kxn and u kxn (x) by solving the 1D problem where we introduced the 1D Hartree potential and is the electron density profile. To obtain Eq. (5), we have employed the prescription and applied the condition of charge neutrality to eliminate x-independent terms. Now, moving to Fourier space, we transform Eq. (4) into a linear system of equations, where G and G are reciprocal lattice vectors (i.e., multiples of 2π/a). This equation involves the Fourier coefficients of different quantities, defined through the relations In particular, from the stepwise profile of the image potential (see above), we find Likewise, from Eq. (5), the coefficients of the Hartree potential reduce to

Supplementary Note 2. QUANTIFICATION OF THE IMAGE ENERGY
We intend to quantify the influence of the image interaction on the electronic behavior in the semiconductor considering the different parameters that define the system. We start from the density n = k 2 F /2π and the kinetic energy E kin = L 2 π 2 n 2 /2m * of a 2DEG. Following a density functional theory approach in the local-density approximation, we write the total energy as a functional of the electronic density n(R): where ∆n(R) = n(R) − n 0 . The ground-state density in the many-electron system is then obtained by minimizing Eq. (7), subject the constraint d 2 R ∆n(R) = 0. By introducing a Lagrange multiplier λ and imposing the vanishing of the functional derivative with respect to n (i.e., δ n E[n] = λ), we find which, transforming all quantities to reciprocal space as in Sec. (Supplementary Note 1), can be written as Now, as a crude approximation, we assume that ∆n(x) has the same periodicity and shape as V im (x), oscillating between the values n 1 and −n 1 for b = a/2 so that the average electron density is conserved. By specifying Eq. (8) at two different points 0 < x 1 < b and b < x 2 < a, and then subtracting the two resulting equations, we find the ratio where A = G =0 (π 2 /4a|G| 2 )K G (e iGx1 − e iGx2 ), K G = ∆n G /n 1 = (2i/aG)(e −iGb − 1), and we use the coefficients V C = e 2 /a and k BZ = π/a defined in the main text. We have verified that A evolves in the (0, 1) interval as the values of x 1 < x 2 are varied. For instance, if x 1 = a/4 and x 2 = 3a/4, we obtain A = ∞ n=0 (−1) n /(2n + 1) 2 ∼ 0.91. We are interested in finding the ratio of the kinetic energy to the image potential energy, which in this approximation becomes and finally, for small perturbations (|n 1 | n 0 ), it reduces to The second fraction in the right-hand side of this equation can be neglected under the conditions investigated in the main text (i.e., for V C /E F 0 /(k BZ /k F 0 ) 2 1), so the influence of the image potential on the material is simply quantified through the parameter V 0 /E F 0 .

Supplementary Note 3. PROBING THE OPTICAL RESPONSE THROUGH EELS
The optical response of the Q-phase materials under consideration can be probed through electron energy-loss spectroscopy (EELS). We follow the general methods discussed elsewhere 3 to calculate the loss probability for an electron moving with constant velocity v parallel to the semiconductor and oriented along in-plane directions either parallel or perpendicular to the ribbons. The electron is taken to be moving in vacuum on the side of the semiconductor that is not occupied by the ribbon structure used to produce the image potential (see insets in Supplementary Figure 1).
Adopting the electrostatic limit, the EELS probability Γ EELS (ω) is directly obtained from the screened Coulomb interaction W (r, r , ω), which describes the potential produced at r by a unit point charge placed at r and oscillating with frequency ω. More precisely 3 , where r 0 describes the electron position at time t = 0, and z 0 is the electron-surface separation. Now, from the analysis presented in the main text, defining G = Gx with G = 2πm/a and m running over integer numbers, we consider points z, z > 0 above the semiconductor surface and express the screened interaction in terms of the Fourier components of the Fresnel reflection coefficient for p polarization: Incidentally, we only write the induced part of the interaction because the direct Coulomb term does not contribute to Eq. (11).  (1/a) a 0 dx 0 e i(G−G )x0 = δ GG , we find where Q 1 = (q x + G) 2 + ω 2 /v 2 , L is the length of the electron trajectory, and the ⊥ superscript refers to the fact that the beam is moving perpendicularly with respect to the direction of the array periodicity. Likewise, for an electron traveling along the transverse ribbon direction x (i.e., parallel to the array periodicity), we can take r 0 = (0, 0, z 0 ) and insert Eq. (12) into Eq. (11) again to readily obtain where Q 2 = ω 2 /v 2 + q 2 y ,q x = ω/v −G, andG is the only lattice vector satisfying the condition |q x | < π/a. Obviously,G may depend on ω/v, although we expect to have ω/v π/a, and therefore G = 0. In such scenario, Eq.
(14) coincides with the electrostatic limit of the probability obtained for an electron moving parallel to the surface of a photonic crystal 4 . In Supplementary Figure 1, we present EELS spectra calculated by using Eqs. (13) and (14) for different strengths of the image interaction V 0 down to the 2DEG limit V 0 = 0 (dashed curves). Strong deviations are observed in the number and positions of the spectral features inherited from the dispersion relations presented in Fig. 2 in the main text. The in-plane anisotropy of the designed Q-phase material gives rise to substantially different profiles for the two electron beam orientations under consideration.